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Zbl 0971.90062
Solodov, M.V.; Svaiter, B.F.
Forcing strong convergence of proximal point iterations in a Hilbert space. (English)
[J] Math. Program. 87, No.1 (A), 189-202 (2000). ISSN 0025-5610; ISSN 1436-4646/e

Summary: This paper is concerned with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by {\it O. Güler} [SIAM J. Control Optim. 29, 403--419 (1991; Zbl 0737.90047)] that the iterates may fail to converge strongly in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in the authors paper [J. Convex Anal. 6, 59--70 (1999; Zbl 0961.90128)]. Strong convergence is foreed by combining proximal point iterations with simple projection steps onto intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two uniknowns.

MSC 2000:: *90C25 Convex programming
46N10 Appl. of functional analysis in optimization and math. programming
65K05 Mathematical programming (numerical methods)

Keywords: proximal point algorithm; Hilbert spaces; weak convergence; strong convergence

Citations: Zbl 0737.90047; Zbl 0961.90128

Cited in: Zbl 1250.47062 Zbl 1132.47051 Zbl 1129.49005 Zbl 1044.90089 Zbl 1101.90083 Zbl 1050.47049

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